Restricted Gauge Theory Formalism and Unimodular Gravity

We develop a Lagrangian quantization formalism for a class of theories obtained by the restriction of the configuration space of gauge fields from a wider (parental) gauge theory. This formalism is based on application of the Batalin-Vilkovisky technique for quantization of theories with linearly dependent generators, their linear dependence originating from a special type of projection from the originally irreducible gauge generators of the parental theory. The algebra of these projected generators is shown to be closed for parental gauge algebras closed off shell. We demonstrate that new physics of the restricted theory, as compared to its parental theory, is associated with the rank deficiency of a special gauge-restriction operator reflecting the gauge transformations of the restriction constraints functions -- this distinguishes the restricted theory from its partial gauge fixing. As a byproduct of this technique a workable algorithm for the one-loop effective action in generic first-stage reducible theory was constructed, along with the explicit set of tree-level Ward identities for gauge field, ghost, and ghosts-for-ghosts propagators. The formalism is applied to unimodular gravity theory, and its one-loop effective action is obtained in terms of functional determinants of minimal second-order operators, calculable on generic backgrounds by Schwinger-DeWitt technique of local curvature expansion. The result is shown to be equivalent to Einstein gravity theory with a cosmological term up to a special contribution of the global degree of freedom associated with the variable value of the cosmological constant. The role of this degree of freedom in a special duality relation between Einstein theory and unimodular gravity is briefly discussed.

IV. One-loop effective action of restricted gauge theory 13 The purpose of this paper is to discuss an interesting property that arises when one restricts a configuration space of gauge theory.This is the origin of a reducible gauge structure characterized by linear dependence of the generators of gauge transformations.Both mechanisms -reduction of the configuration space and reducibility of gauge symmetry -are widely known and applied in various areas of field theory.In the gravity theory context one of the most important examples of the field space restriction is unimodular gravity (UMG) theory which was suggested by Einstein soon after the invention of general relativity [1].Much later it was considered in the context of particle physics [2], in the context of a spacetime covariant formulation [3], as a problem of time and the cosmological constant problem [4], and then applied within the dark energy paradigm [5] with the emphasis on purely technical issues of perturbation theory, etc. Extension beyond the unimodular constraint on the metric of spacetime -the so-called generalized unimodular gravity [6] -also turned out to be rather fruitful from the viewpoint of the generation of viable dark energy and inflation scenarios [7,8].
From a field-theoretical point of view UMG is interesting because it is generally observationally equivalent to general relativity (see recent review [9]), but even semiclassically raises the issues of such an equivalence [10] depending on the subtleties of local physics vs global behavior encoded in boundary conditions, finiteness of the spacetime volume, thermodynamical setup in gravity theory [11], etc. Unimodular gravity is an interesting object of group theoretical analysis [12] and quantization [13][14][15][16][17], especially regarding its relation to quantization of Einstein theory [18] as a sort of a parental theory whose field space reduction leads to the unimodular gravity model.
On the other hand, it has been conjectured [6] that the formalism of quantized unimodular and generalized unimodular gravity theories can be developed along the lines of gauge theory quantization in models with linearly dependent (or reducible) generators [19,20].This observation follows from a simple fact that the restriction of the configuration space of metrics to the subspace of metrics with a unit determinant results in the relevant reduction in the space of diffeomorphism invariance parameters, which can be attained by a projection procedure.Then, if one wants (for the sake of retaining manifest covariance) to describe the restricted theory in terms of the parental one, this projection applied to the original gauge generators makes them linearly dependent.
To the best of our knowledge this relation between the reducible nature of gauge generators and the reduction of the field space has not yet been fully exploited even in the unimodular gravity context, as well as its extension to more or less general gauge theories.The studies of theories with restrictions on gauge transformations, the theories with the so-called unfree gauge parameters, were recently intensively conducted in a series of publications [21,22] with the purpose of constructing the BFV (Batalin-Fradkin-Vilkovisky) formalism transcending these restrictions to the ghost sector of the model and starting from the first principles of Hamiltonian formalism and canonical quantization.Here we choose a somewhat different approach (closer to the Lagrangian formalism) and try to develop the idea of [6] for a generic parental gauge field theory with the closed algebra of its irreducible generators becoming linearly dependent in the course of a projection induced by the restriction of the original configuration space.
Our method is based on the explicit construction of a projector operator with which the generators of the original parental theory are projected onto the generators of the residual symmetries of the restricted theory.Linear dependence of the latter is then treated by the quantization method of Batalin and Vilkovisky (BV) for reducible gauge theories [19,20] -a brief review of this formalism with all necessary notations and definitions is presented in Sec.II A. As a by-product of using BV formalism we develop in much detail the gauge-fixing procedure for gauge theory of the first-stage reducibility.Quite interestingly, despite numerous applications of this method, current literature does not present a workable algorithm even for the one-loop approximation in generic reducible gauge theories.Implicitly this algorithm is, of course, contained in the pioneering works [19,20], but its concrete realization with all the details of Ward identities providing gauge independence of the on-shell effective action is still missing.We close this omission and suggest the recipe for all the elements of Feynman diagrammatic technique and the relevant algorithm of the one-loop effective action in the above class of theories at the level of presentation characteristic of a folklore use of the wellknown Faddeev-Popov prescription.
The gauge-fixing algorithm here turns out to be more involved than in a conventional Faddeev-Popov formalism for irreducible theories, because it includes gauge fixing for original gauge fields, gauge fixing for ghosts and the corresponding ghosts for ghosts.Even for firststage reducible theories (when there is no higher-order zero vectors for reducibility generators -zero vectors of original gauge generators) this gauge-fixing procedure involves, apart from the usual gauge conditions, at least three extra gauge-fixing elements and the construction of a special projector in the gauge-breaking term of the theory.The main result for the one-loop effective action of gauge theories of the first-stage reducibility is assembled in the sequence of Eqs.(2.47)-(2.51) of Sec.II C.
Then in Sec.III we explain how a reducible gauge structure with linearly-dependent generators arises in theories with restricted configuration space of gauge fields.In particular, we raise the question why and when constraining this space can be regarded merely as a partial gauge fixing, or on the contrary, forming a new physical theory inequivalent to the parental one.It turns out that the answer is based on the rank of a special gauge-restriction operator.The rank deficiency of this operator, or the presence of its nontrivial zero modes, signifies new physics of a restricted theory as compared to the parental model whose field space is subject to restriction.This gauge-restriction operator allows one to construct the projector on the space of residual gauge symmetries.The projected gauge generators form the algebra which is closed as long as the original parental theory has a closed algebra both on and off shell.
The zero vectors of this projector become first-stage reducibility generators of the BV method of treatment of the restricted theory, and they represent a free element of the formalism, whose choice is limited only by a certain rank restriction condition.It turns out, however, that their special normalization fully provides onshell independence of the one-loop effective action of this choice, whereas the normalization of the gauge generators of the parental theory can be fixed by matching it with the canonical formalism and canonical quantization of parental theory.
The one-loop effective action for generic restricted theory is discussed in Sec.IV A. Discussion of correct normalization of the gauge generators in a covariant formalism is the subject of Secs.IV B and IV C. In Sec.IV D we discuss the factor in one-loop effective action which distinguishes restricted and parental theories at the quantum level.
In Sec.V we apply the above formalism to the unimodular gravity theory treated as descending from its parental version -Einstein general relativity.We construct all the elements of its gauge-fixing procedure in the background covariant gauge [23] with the background covariance property extended to all the objects of the Feynman diagrammatic technique.Then, according to the general algorithm, we build on shell the one-loop effective action of the UMG theory.This confirms conclusions of [15] which were attained, however, with certain assumptions on the treatment of the group of diffeomorphisms volume factored out of the partition function.No such assumptions are needed in our approach which is fully determined by the requirement of gauge independence of the on-shell physical results.
Moreover, our approach allows us to disentangle the one-loop contribution of a global zero mode responsible for the "new" physics contained in the UMG theory as compared to general relativity.The UMG one-loop effective action coincides with that of the Einstein theory having a cosmological constant modulo a special contribution of the global degree of freedom associated with the variable value of this constant.The resulting expression for the effective action, which is usually presented in terms of functional determinants on irreducible (transverse and transverse-traceless) subspaces of the full tensor and vector field space, is built in terms of the functional determinants of the minimal operators.These determinants admit for their calculation the application of the Schwinger-DeWitt curvature expansion [24].This property is important for calculations on generic back-grounds not exhausted by homogeneous spaces.
In the Conclusion we summarize the results and briefly discuss the role of a global extra degree of freedom in the UMG theory as a cosmological constant.In particular, we show how this reveals the duality of UMG and Einstein theories analogous to the transition between statistical ensembles with fixed observables related by the Laplace duality transformations.In the appendixes we discuss the Moore-Penrose construction of inverse operators for degenerate operators [25], as well as the derivation of the gauge algebra of the projected generators and some technical details of determinant relations needed for the construction of the generator basis.

II. EFFECTIVE ACTION FOR THEORIES WITH FIRST-STAGE REDUCIBLE GAUGE GENERATORS
In this section we briefly review Batalin-Vilkovisky (BV) formalism for the first-stage reducible gauge theories.Throughout this section (and in the most of other sections unless otherwise indicated) we use DeWitt condensed notations.All indices are of condensed nature and combine discrete bundle and continuous spacetimepoint labels.Two-index quantities are two-point kernels and summation over contracted indices implies the corresponding spacetime integration.Ranges of condensed indices reflect the continuum of spacetime points and the discrete spin-tensor labels of fields.The ranks of twopoint kernels of linear operators and forms thus refer to their functional space dimensionality.

A. Batalin-Vilkovisky formalism for reducible gauge theories
Consider a generic gauge theory of fields φ i (with the range of field indices i formally denoted in what follows by n).Let it be described by the action S[ φ ] which is invariant under the local gauge transformations with generators R i α = R i α (φ) and infinitesimal gauge parameters ξ α (the range of gauge indices α being denoted by Here, and in what follows, comma denotes a functional derivative with respect to the relevant field variable.The algebra of gauge generators begins with the relations -corollaries of the Jacobi identity for a double commutator of gauge transformations with the structure functions For closed algebras (in contrast to so-called open algebras) E ij αβ (φ) and higherorder structure functions of the gauge algebra vanish [26,27].
The theory is reducible when its gauge generators are linearly dependent on shell, that is on solutions of classical equations S ,i = 0, so that there exist m 1 reducibility generators Z α a = Z α a (φ) -right zero-eigenvalue eigenvectors of the original generators R i α parametrized by indices a of range m 1 .This implies relations with some coefficient functions B ij a (φ) = −B ji a (φ).For the first-stage reducible theories, which we consider in this section, the generators Z α a form on shell a complete independent set, whereas for higher-order stages of reducibility they also become linearly dependent with higher-order reducibility generators and so on.

Master action
Batalin-Vilkovisky quantization of gauge theories starts with the construction of master action S BV [ Φ, Φ * ] which is a certain extension of the original action classical S[ φ ] onto the configuration space of fields and antifields (Φ I , Φ * I ).This even-dimensional space is Z-graded with respect to the so-called ghost number gh (Φ * I ) = − gh (Φ I ) − 1 and Z 2 -graded with respect to their Grassmann parity ǫ, ǫ(Φ * I ) = ǫ(Φ I ) + 1. Original fields φ i become a part of extended fields set Φ I and carry zero ghost number.To avoid messy sign factors we assume that all φ i are bosonic (even) fields, ǫ(φ i ) = 0, and Grassmann parity of ghost fields and antifields coincides with Z 2 parity of their ghost number (those with even ghost numbers are commuting fields, those with odd ghosts numberanticommuting).
Extension from ) in terms of the antibracket (G, H) [28] which is defined for any two functionals G and H of fields and antifields.Here (r) and (l) label, respectively, right and left derivatives.
The solution S BV [ Φ, Φ * ] is obtained as a perturbative expansion in powers of ghost fields and antifields.To make this solution proper and allow one to perform a further gauge-fixing procedure one should specify a BVextended configuration space and impose certain initial conditions to (2.5) which should explicitly encode the gauge structure of the original action S[ φ ].
These generators, however, form a reducible set because the second variational derivative of (2.6) shows their N -reducibility on shell, Thus, the master action S BV possess N on-shell independent symmetries -half the dimension of the BV configuration space (Φ I , Φ * I ). 1 Convenient gauge fixing of the master action gauge symmetry (2.7) consists in expressing antifields in terms of fields via N gauge conditions [19,[26][27][28][29][30]] It is parametrized by a single functional of fields -gauge fermion Ψ [ Φ ] of ghost number −1 and leads to gaugefixed action which is functional of only BV fields Φ I .For the correctly chosen gauge fermion the gaugefixed action (2.9) appears to be a nondegenerate sdet [ δ (l) δ (r) S Ψ /δΦ I δΦ J ] δSΨ /δΦ=0 = 0 and yields a perturbatively consistent path integral for the generating functional which modulo the contribution of local measure reads (2.10) The master equation for S BV [ Φ, Φ * ] (2.5) provides the corner stone of gauge theory quantization -independence of the on-shell generating functional of the choice of gauge fermion.Under the gauge fermion change Ψ → Ψ + ∆Ψ , in view of integration by parts the change in Z reads where extra terms vanish in view of Grassmann symmetry properties.This expression equals zero because 1 Nilpotence of 2N × 2N matrix implies that its rank is at most N .Considering a proper solution implies that rank of the Hessian δ l δ r S BV /δΦ A BV δΦ B BV equals N .Otherwise the set of Noether identities (2.7) is incomplete and there are more then N gauge generators since there are more then N zero modes of the Hessian.
the first term vanishes in view of the master equation δ (r) S BV /δΦ * I δ (l) S BV /δΦ I = 1 2 (S BV , S BV ), while the second term in local theories is proportional to ∂...∂ δ(0) and is supposed to be either canceled by a local measure or killed by dimensional regularization.The local measure is not rigorously available within the BV formalism whose incompleteness can be disregarded for theories with local gauge algebra and within the class of local gauge fermions.This measure can be attained within the canonical BFV quantization formalism, which will be used below to justify the application of the BV method for restricted gauge theories in which locality of their generators will be spoiled by generically nonlocal projectors.

Minimal and nonminimal proper solutions
Construction of BV master action begins with finding a proper solution of master equation (2.5) on the minimal sector of fields and antifields.For the first-stage reducible theories (2.2), (2.4) this requires introduction of ghosts C α , gh (C α ) = 1, the ghosts for ghosts C a , gh (C a ) = 2, and antifields to all fields and ghosts [19,26,27], 12) The minimal proper solution S min [ Φ min , Φ * min ] of the master equation can be represented as the series in powers of antifields where S = S[ φ ] is the action of the initial gauge theory, two terms bilinear in ghost fields and antifields serve as initial conditions for the master equation which guaranty that such a solution of master equation is proper, R i α and Z α a are gauge and reducibility generators (2.2),(2.4),dots stand for higher-order terms in powers of ghost fields and antifields.All higher-order terms can be found within the iterative procedure of solving the master equation which leads to [26,27] where the coefficient functions of higher-order terms originate from higher-order structure functions of gauge algebra and the dots denote terms with antifields of the total ghost number −3 and lower.Meaningful gauge fixing of the BV master action requires extension of the proper solution to nonminimal configuration space.Motivation for this is that gauge fixing (2.8) is performed through properly constructed gauge fermion Ψ [ Φ ] -the functional of only the BV fields Φ I .But the gauge fermion has a negative ghost number −1 and thus cannot be constructed as a functional of the minimal fields φ i , C α , C a with nonnegative ghost numbers.Therefore, the set of fields should be further extended to contain auxiliary ghosts with negative ghost numbers.
Standard scheme for first-stage reducible theories presumes introducing auxiliary ghosts Cα , Ca , C ′a , their partners π α , π a , π ′a with the higher ghost number and the corresponding set of antifields [26,27], so that nonminimal BV configuration space reads Here the fields Cα , Ca are often referred to as antighosts, C ′a -extraghosts, π α , π a , π ′a -Lagrange multiplier or Nakanishi-Lautrup fields.
Ghost numbers of the nonminimal configuration space of the first-stage reducible gauge theory are listed in Table I, where antifields π * α , π * a , π ′ * a are omitted since they do not appear in the BV procedure.
The master action (2.15) obviously satisfies the master equation (2.5) on the nonminimal configuration space since auxiliary and minimal sectors are so far decoupled (these sectors will be mixed after gauge fixing) so that (S min + S aux , S min + S aux ) = (S min , S min ) + (S aux , S aux ), and by construction S min and S aux separately nilpotent in antibracket.

Choice of a gauge fermion for the Gaussian gauge fixing
Grading restriction gh (Ψ ) = −1 on gauge fermion Ψ [ Φ ] admits the following form at most quadratic in ghosts and auxiliary fields The latter group of terms linear in π α , π a , π ′a allows us to implement Gaussian (Faddeev-Popov) type of gauge fixing when κ αβ , ρ a b are nondegenerate In generic theories it is convenient to choose φindependent coefficient functions κ αβ , ρ a b .Such a choice guarantees the possibility to integrate out auxiliary fields π α , π a , π ′a to obtain Faddeev-Popov representation for the generating functional.
Here we take into account that for φ-dependent functions κ αβ (φ), ρ a b (φ) terms linear in companion fields π α , π a , π ′a will arise in gauge-fixed minimal master action S min Ψ instead of each antifield, φ * i = δΨ/δφ i = 1 2 C α (δκ αβ /δφ i ) π β + ... .Therefore, for open algebras and algebras with higher-structure functions, whose S min contains second and higher powers of antifields, π α , π a , π ′a dependence of S min Ψ becomes more than quadratic, which does not allow us to integrate them out via Gaussian integration.
After the gauge fixing (2.9), Φ * → δΨ/δΦ, with the gauge fermion (2.17) one gets the nonminimal gaugefixed action (2.9) which according to (2.15) where S = S[ φ ] is the action of the initial gauge theory and dots hide terms more then quadratic in ghost fields with nonzero ghost numbers. 2 We introduce the combination which depends on fields φ i , C ′a with zero ghost number.
All the dependence on C ′ in the gauge-fixed action S Ψ is now hidden within X α and its variational derivative For φ-independent matrices κ αβ and ρ a b -the case which we consider in what follows -Gaussian integration over the fields π α , π a , π ′a in the generating functional (2.10) gives where Φ red is the reduced nonminimal set of fields Φ red = (φ i , C ′a , C α , Cα , C a , Ca ), cf.Eq. (2.14), and S F P [ Φ red ] is the corresponding Faddeev-Popov reduced gauge-fixed action Here we have introduced the matrices κ αβ , ρ −1 a b inverse respectively to κ βα , ρ b a .Dots denote terms which are more than quadratic in ghost fields C α , Cα , C a , Ca .Such terms are irrelevant in the one-loop approximation.
The obtained reduced action is nondegenerate in the sense that its Hessian at the stationary points of the classical BV action represents the nondegenerate operator sdet δ (l) δ (r) S F P δΦ red δΦ red δS F P /δΦ red =0 = 0 (2.25) for the relevant set of boundary conditions.This property is inherited from the nondegeneracy of the unreduced gauge-fixed action S Ψ [ Φ ] (with appropriately chosen gauge fermion), in which auxiliary nondynamical variables were integrated out (being expressed from their own equations of motion).For nondegenerate gaugefixing matrices κ βα , ρ b a fields π α , π a , π ′a form the set of such auxiliary variables.Now we will analyze nondegeneracy in various sectors of the reduced configuration space.

B. Stationary point of the gauge-fixed master action
The simplest is the ghost sector consisting of the fields C α , Cα , C a , Ca (the field C ′a with a zero ghost number does not belong to this sector even though it is usually called the extraghost [19] -rather it belongs to zero ghost number sector where it plays a special role in gauge-fixing procedure for φ i ).The variational equations for the fields C α , Cα , C a , Ca obviously make them vanishing under zero boundary conditions provided the kernels of their bilinear forms in gauge-fixed action (2.24) represent invertible operators having well defined Green's functions The interpretation of these operators and their properties is obvious.The first term of the ghost operator F α β is degenerate because of the reducibility of gauge generators, so the conventional Faddeev-Popov ghosts C α , Cα themselves become gauge fields with the local symmetry induced by the reducibility generators Z α b .The second term of F α β in (2.27) plays the role of gauge breaking term for this symmetry, and its effect is the invertibility of F α β .Correspondingly, F a b is the Faddeev-Popov operator of new ghosts C a , Ca for original ghosts C α , Cα treated as gauge fields, that is why the fields C a , Ca bear the name of ghosts for ghosts.
The situation is trickier in the zero ghost number sector of fields φ i , C ′a .Their equations of motion read ) Contracting them respectively with R i γ and ρ −1 b a ω a γ and subtracting from one another we obtain, on account of the Noether identity S ,i R i γ = 0, the following on-shell relation: ) which allows one in view of invertibility of F α γ and κ αβ to rewrite the equations of motion in the zero ghost number sector as This can be interpreted as equations of motion for φ i supplied by the set of gauge conditions X α = 0.The latter look overcomplete because the number of independent equations of motion S ,i = 0 on n variables φ i is n−m 0 +m 1 , whereas the number of gauges is m 0 .The m 1 mismatch is, however, corrected by the m 1 extra ghost fields C ′a .This goes as follows.
Contracting the rank m 0 nondegenerate operator F α β with the rank m 1 full-rank generator Z β c one gets the full-rank condition, rank (σ This, in particular, guarantees that all C ′a are expressible in terms of φ i from the equations X α = 0. By introducing an arbitrary matrix s a α (φ i ) such that det (s a α σ β b ) = 0 one can express the solution for C ′a in terms of φ i from Eq. (2.33) where (sσ) −1 a b is the inverse of (sσ) b c = s b γ σ γ a .On the substitution of this solution in (2.33) the rest of relations constitute m 0 − m 1 independent gauge conditions which explains why Eq. (2.35) comprises m 0 − m 1 gauge conditions rather than m 0 conditions.It is remarkable that despite the presence of an auxiliary element -an arbitrary matrix s b β of rank m 1 -these gauge conditions and expressions for extraghosts (2.34) are s-independent.This easily follows on shell from the variational equation (2.38) Finally, consider the on-shell Hessian of the action in the zero ghost number sector of fields Φ 0 = (φ i , C ′a ) which is actually the Hessian of S[ φ ] − 1 2 X α κ αβ X β .Bearing in mind the on-shell value of X α = 0 we get the block-matrix operator In view of invertibility of κ αβ one can introduce the nondegenerate m 1 × m 1 operator κ ab and its inverse (2.40) Then one can factorize the determinant of (2.39) as the product of determinants of two operators det where the new operator F ij is obviously the gauge-fixed inverse propagator of fields φ i in which the gauge-fixing term is built in terms of the gauge matrices X α ,i and the gauge-fixing matrix Π αβ .The latter, however, does not coincide with the original matrix κ αβ in the gauge fermion (2.17), but rather converted to the projector form with the following properties: This projection is fully consistent with the fact that the rank n − m 0 + m 1 of S ,ij should be raised up to n by adding nonzero eigenvalues not in the m 0 -dimensional subspace, but in the (m 0 − m 1 )-dimensional one.In particular, with the projector (2.43) one can rewrite onshell gauge conditions (2.35) on fields φ i as Π αβ χ β = 0 .The equivalence of this to (2.35) is established by choosing s a β = m ab σ α b κ αβ and noting that the projector δ α β − σ α a (sσ) −1 a b s b β does not depend on a nondegenerate matrix factor m ab .

C. One-loop contribution to the generating functional
From (2.23) the one-loop contribution to the generating functional (its preexponential factor) reads (2.45) where the Hessian in the sector of reduced fields has in virtue of (2.24) the following block matrix structure (2.46) Block in the left column and upper row here corresponds to the zero ghost number sector Φ 0 = (φ i , C ′b ), the middle column and row block corresponds to the odd sector (C β , Cβ ) of gauge ghost and antighost fields, right bottom block corresponds to the even sector of ghost-for-ghost and antighost-for-ghosts fields (C b , Cb ).
In virtue of the factorization property in the reduced fields block (2.41) the calculation of the full superdeterminant finally gives the one-loop contribution to the generating functional of the first-stage reducible gauge theory Let us assemble together all numerous ingredients of this expression which were introduced above in the course of derivation of this formula.Here the inverse propagator of the original gauge field F ij (2.42), the ghost operator F α β (2.27), and the ghosts-for-ghosts operator F a b (2.28) are, respectively, (2.50) Gauge field and ghost operators (2.48), (2.49) are both gauge fixed with the aid of gauge conditions matrices X α ,i , σ α a , and ω b β and the gauge-fixing matrices Π αβ and ρ −1 a b -kernels of bilinear terms in (X α ,i , σ α a , ω b β ).Whereas these matrices of gauge conditions for ghosts σ α a and ω b β are a part of the originally chosen gauge fermion (2.17), a similar matrix in the gauge fields sector is a special projector Π αβ (2.43), Π αβ κ βγ Π γ δ = Π α δ (2.44), on the direction in the space of gauge indices orthogonal to the "vielbein" σ α a , σ α a Π αβ = Π αβ σ β b = 0.This type of "orthogonality" is determined with respect to the symmetric metric κ αβ , which is inverse to κ βα originally introduced in the gauge fermion (2.17).The projection of this metric onto the space of reducible generator indices a with the aid of the vielbein σ α a gives rise to the gauge-fixing matrix κ ab in the space of these indices.The determinants of all gauge-fixing matrices κ αβ , ρ a b and κ ab appropriately enter the final algorithm for the one-loop generating functional (2.47).
Ranks of all the above gauge matrices are maximal and determined by the range of their indices, and the main criterion of their choice is the invertibility of the full set of gauge and ghost operators (2.48)-(2.50).
The final important comment is the definition of the on-shell condition for the obtained algorithm (2.47).All ghosts, ghosts for ghosts and their antighost fields are zero on shell, C α = Cα = C a = Ca = 0 (2.26).The exception is the only "non-classical" field C ′ a which has a zero ghost number, though originally it was called the extraghost [19].On shell it is generically nonvanishing and is given by the expression (2.34), so that the gauge conditions matrix equals and coincides with χ α ,i only for φ-independent σ α c or vanishing on shell s b β χ β .For the original gauge field φ i onshell restriction and gauge conditions (2.32)-(2.33)equivalently read S ,i = 0, Π αβ χ β = 0 (note that the original gauge functions χ α (φ) introduced in the gauge fermion (2.17) does not necessarily vanish -only their projection vanishes on shell).

D. Ward identities and gauge independence of the effective action
The one-loop effective action corresponding to (2.47) reads and includes gauge field, ghost, ghost-for-ghost contributions and the contribution of three gauge-fixing matrices.On shell this expression should not depend on all gauge-fixing entities χ α (φ), σ α a (φ), ω a α (φ), κ αβ , ρ a b as it is dictated by the general BV theory.It is worth checking this property and revealing the details of the perturbative mechanism of such gauge independence.In the one-loop approximation this mechanism is based on the on-shell Ward identities for all tree-level propagators of the theory: Ward identity for the Green's function of the gauge field operator F ij follows from the sequence of on-shell relations 53) where we used the fact that on shell R k γ is a zero vector of S ,jk since S ,jk R j γ = −S ,j R j γ,k , and whence it follows the effective action independence on the choice of gauge matrix X α ,i and the gauge condition it follows that the Ward identity relating the ghost and ghosts-for-ghosts propagators is

56) and the effective action turns out to be on
57) The dependence of Γ 1−loop on σ α a involves three terms of (2.52) the sum of which also turns out to be zero on shell To prove this one should use in the variation of the operator (2.48) the variation of the projector , and the corollaries of (2.54) and (2.56), ) (2.61) The κ αβ -variation of the effective action also vanishes on shell because of the variations δ Finally, on-shell independence of the gauge-fixing matrix ρ a b follows by direct variation and the use of the corollary (2.61) of the Ward identity (2.56) for the ghost propagator, (2.63)

III. REDUCIBLE GAUGE STRUCTURE OF RESTRICTED GAUGE THEORIES
Reducible structure of a gauge theory can be induced by the procedure of restriction of the originally irreducible gauge theory.To see this consider a generic gauge theory with the action Ŝ Irreducibility of the generators RI α implies that the rank of their matrices coincides with the range of the indices α which in its turn is lower than the range of the indices I enumerating the original gauge fields ϕ I , rank RI α = range α = m 0 < n = range I.We will call such a theory the parental one.
Restricted gauge theory, originating from the parental one, is the theory whose configuration space variables are kinematically constrained by the equations where we will consider the functions θ a (ϕ I ) to be functionally independent, that is characterized by the fullrank condition of their gradient matrix, For simplicity we assume that these functions are either ultralocal (algebraic) in spacetime or the surface of these constraint functions can be parametrized in terms of local independent fields which will be denoted in what follows by φ i .

A. Two representations of a restricted theory
Such a restricted theory can be described in two equivalent ways.One way is to represent it in terms of the Lagrange multipliers action whose classical equations of motion obtained by varying both its gauge fields ϕ I and Lagrange multipliers λ a read Another representation is the reduced theory, when one solves first the constraints (3.2) with respect to ϕ I as functions of the reduced set of fields φ i and formulates the theory in terms of the reduced action S red [ φ ].The latter is obtained by substituting in the parental theory action the functions e I (φ) of embedding the φ i -subspace into the space of original gauge fields ϕ I , θ a e I (φ) ≡ 0, (3.6) The equations of motion in both formulations are obviously equivalent 3 because the contraction of the first of Eqs.(3.5) where we took into account that in view of (3.6) the covector θ a ,I is orthogonal to the surface of constraints θ a = 0, An important question is the relation of the original parental theory and the restricted one from the point of view their physical equivalence.Solutions of the system of equations (3.5) are obviously inequivalent to those of the parental theory, Ŝ,I = 0, when on shell the Lagrange multipliers λ a are nonvanishing.Note that in view of linearity of the λ-action (3.4) in λ a the variational equations with respect to Lagrange multipliers do not allow one to express them from the full set of equations of motion.There is another way to write down their equations of motion by contracting the first set of equations (3.5) with the parental theory generators RI α and use the Noether identities (3.1).Then we get λ a Q a α = 0, (3.10) 3 Classical equivalence of two theories can be defined by equivalence of spaces of solutions of their equations of motion.In particular it can be established by the local correspondence (3.8) and the statement that reduced fields do not carry additional degrees of freedom.This statement means that the above assumptions on θ a (ϕ) imply the existence of such reducibility, that is the possibility of excluding auxiliary variables in terms of the reduced ones.
where Q a α = Q a α (ϕ) is the gauge-restriction operator which will be very important in what follows, This equation for λ a has a unique solution λ a = 0 only when the rank of Q a α is maximal, that is it coincides with the range of the index a enumerating the constraint functions θ a .In this case the meaning of the constraints θ a (ϕ) = 0 is nothing but a partial gauge fixing of gauge invariance of the parental theory (range a = m 1 < m 0 = range α).
On the contrary, when the rank of gauge-restriction operator then m 2 Lagrange multipliers can be freely specified, the rest m 1 − m 2 of them being fixed as unique functions of the former free ones.This implies, in particular, the existence of m 2 gauge invariants of the parental theory constrained to be zero in restricted theory.To see this, note that the rank deficiency implies that operator and this, according to the definition of this operator, implies the existence of m 2 parental gauge invariants For local in spacetime restriction functions 4 θ a this is a local expression for Lagrange multipliers in terms of Ŝ,I -the left-hand side of equations of motion of the parental theory.Resolving Lagrange multipliers λ a in restricted equations of motion (3.5) leads to the projected set of equations, Ŝ,J (δ J I − θ J a θ a ,I ) = 0, (3.17) equivalent to Ŝ,J e I ,i = 0.The projector (δ J I − θ J a θ a ,I ) enforces only a part of parental equations of motion Ŝ,I "tangential" to restriction surface θ a = 0. Which is equivalent to (3.8) and reinstates the observation that along tangential directions equations of motion for restricted theory coincides with that for parental theory.
On the contrary, projected on complementary ("normal") directions, equations of motion for these two theories differ.Contraction of the parental equations Ŝ,I = 0 and restricted theory equations Ŝ,I − λ a θ a ,I = 0 with θ I a gives correspondingly Ŝ,I θ I a = 0 and Ŝ,I θ I a = λ a .The normal subset of the parental equations in addition to equations (3.17) further restricts classical configurations of fields ϕ I in parental theory.While the normal subset of the restricted theory equations merely expresses λ a in terms of ϕ I and thus does not additionally constrain the latter. 5However due to gauge invariance of the parental theory (3.1) not all dynamical restrictions on ϕ I in the right-hand side of (3.16) are effectively removed.The linear dependence Ŝ,I RI α = 0 between Ŝ,I may express a certain part of Ŝ,I θ I a in (3.16) as linear combinations of tangential equations (3.17) which are still frozen to 0 within the restricted equations of motion.Obviously such linear combinations are found by first contracting (3.16) with θ a ,I and then with RI α which leads to λ a Q a α = Ŝ,I θ I a Q a α .The same structure Ŝ,I θ I a Q a α appears in the left-hand side of tangential equations (3.17) after contracting with RI α and thus vanishes on-shell.This reinstates constraints on Lagrange multipliers (3.10), λ a Q a α = 0.

B. Gauge symmetry and reducibility
We assume that the restricted theory does not acquire local gauge symmetries beyond those of the original parental action.On the other hand, not all gauge transformations of the the parental theory δ ξ ϕ I = RI α ξ α generate symmetries of the reduced action (3.4).These symmetries are only those which preserve the constraints so that with the definition (3.11) the allowed gauge transformation parameters ξ α red in the restricted theory should be solutions of the linear equation Q a α ξ α red = 0, i.e. right zero vectors of the matrix Q a α .The subspace of reduced gauge parameters can be obtained by projecting gauge parameters of the parental theory ξ α with a projector T α β which is orthogonal in the space of gauge indices to Q a α , On the other hand, gauge symmetries of the restricted theory can be formulated in the parental field space of ϕ I with free (nonreduced) gauge parameters ξ α .This happens if instead of projecting ξ α we project with respect to the gauge index α the original parental generators RI α .Thus we get the reducible set of gauge generators This formulation is covariant from the viewpoint of the parental gauge theory because the original multiplets of parental field indices I and gauge indices α remain unsplitted, but the price paid for this covariance is the reducibility of the generators R I β .They are indeed reducible because the projector T α β in view of rank deficiency (following from (3.19)) possess right zero vectors k β b which become also the right zero vectors of Despite reducibility an important advantage of such projected representation is that it generates gauge transformations with arbitrary unrestricted gauge parameters ξ α .In the course of subsequent Batalin-Vilkovisky extension of configuration space it will give rise to gauge ghosts equivalent to that of the parental theory.This property is achieved at the cost of reducibility of the projected gauge structure (3.20) and introduction of ghost for ghosts fields.
The projector T α β may be explicitly constructed in terms of the left-and right-kernel bases (3.19),(3.21) Left kernel -Q a α (3.11) -is fixed by the restriction conditions and the choice of parental gauge generators.Thus (3.22) may be considered as the family parametrized by its right kernel basis k β b .The structure of T α β (Q, k), which in what follows will be referred without specification of kernels, is analogous to the set of projectors T α β (s, σ) (2.36) introduced above.
Here we emphasize the property that the matrix (Qk) a b ≡ Q a γ k γ b is not directly invertible.Rather it could have rank equal or lower to that of rank Q a γ = m 1 − m 2 .The rank deficiency is dictated by the requirement of new physics incorporated by the restricted theory (3.12) -physical inequivalence to the parental gauge theory.As discussed in Appendix A, this extra difficulty can be circumvented by the Moore-Penrose construction of the generalized inverse [25], provided the following rank restriction conditions (3.23) This guarantees unambiguous definition of projector (3.22) and its correct rank property, rank Rank restriction requirements (3.23) in particular are guaranteed when k β b is parametrized in terms of nondegenerate two-forms B βα (ϕ) and c ab (ϕ) -metrics in spaces of α-indices and a-indices respectively, which will be used in unimodular gravity calculations in Sec.V.In fact in this case the projector T α β is defined by Q a α and B βα only because the dependence on c ab completely cancels out. 6 The generators (3.20) of gauge transformations in restricted theory 7 are thus far defined in the parental theory field space of ϕ I .They determine the gauge transformations δ red ξ ϕ I = R I β ξ β tangential to the surface of θ a (ϕ) = 0.As any field on this surface can be parametrized by φ i these gauge transformations can be expressed via the gauge transformations of the reduced restricted gauge theory δ red ξ φ i = R i β ξ β , δ red ξ ϕ I = e I ,i δ red ξ φ i , so that the corresponding generators R i β of reduced space gauge fields φ i are related to R I β by obvious pushforward relations Thus, the reduced-space representation of the restricted gauge theory has reducible generators R i β and 6 The "symmetric" form (3.24) has the structure analogous to that of symmetric projector Π αβ (2.43) (with the left index raised) when κ αβ ↔ (B βα ) −1 and σ α b ↔ B αβ Q a β c ab . 7In this section we focus on the gauge transformations (3.20) of fields ϕ I in the restricted theory (3.4).However, the configuration space of the latter also contains fields λa.The Noether identities for the equations of motion (3.5) show that the gauge transformation of Lagrange multipliers should be zero, δ ξ λa ≡ Raαξ α = 0.This is so when the projector T α β (Q, k) (3.22) in the gauge generators (3.20) is constructed with respect to the operator Q a α (3.11) in the whole neighborhood of the restriction surface θ a (ϕ) = 0. This, in particular, implies that Q a α has a constant rank in this neighborhood.We assumed this property and so the gauge symmetry representation their first-stage reducibility generators where Z β a exhaust linear combinations of right zero vectors of the projector T α β and have the form Note that the projectors (3.22) generically are nonlocal depending on structure of restriction functions and gauge generators involved in the construction of gaugerestriction operator Q a α .

C. Gauge algebra of a restricted theory
To proceed further we have to know the gauge algebra of the restricted theory generators.The a priori maximum that can be stated from the first of their relations (3.26) (under certain assumptions of regularity of R i β (φ) [30]) is that they satisfy an open algebra (2.3) with some structure functions and S ,i replaced by S red ,i .Quite remarkably, we will be able to derive this algebra first by expressing via Eq.(3.25) R i β in terms of R I β and using the algebra of the projected (full space) generators (3.20), R I β = RI α T α β .The latter, in its turn, follows from the gauge algebra of the parental theory and turns out to be also closed in the case of closure of the algebra of RI α .Thus, finally we will show that the resulting algebra is also closed as long as we start with the closed algebra of the parental theory.
Expressing R i β in terms of R I β goes with the aid of covariant vectors e i I dual to e J ,j ,i e the last equation here expressing completeness of this basis. 8For fixed e I ,j and θ a ,I there is a freedom of choosing complementary basis elements where square brackets denote pairs of antisymmetrized indices and the second term in the middle part vanishes because for dual e i J and e J ,k (3.29) e i J,j e J ,k = −e i J e J ,kj ≡ −e i J δ 2 e J (φ)/δφ k δφ j and so antisymmetrization in indices kills it.Thus the gauge algebra of R i α directly follows from that of R I α .As shown in Appendix B the commutator of projected generators R I α replicates the algebraic relation of the parental theory.If the latter is prescribed by RI with new structure functions C γ αβ and where Thus, in view of (3.32) for a closed algebra of the parental theory with ÊIJ αβ = 0 one gets a closed gauge algebra of reduced fields representation of the restricted theory with E IJ αβ = 0 and the same structure functions (3.35) These structure functions can be nonlocal and with respect to its lower indices have both transversal and longitudinal nature regarding the projector T α β because of the properties of N γ αβ -tail of the expression (3.35), with the aid of the nondegenerate metric GIJ on the manifold of ϕ I .This induces the metric G ij ≡ e I ,i GIJ e J ,j , G ij ≡ (G ji ) −1 , on the reduced φ i -space -the surface of θ a (ϕ) = 0, and its inverse G IJ ≡ (GJI ) −1 induces the metric in the directions normal to this surface, G ab ≡ θ a ,I G IJ θ b ,J , G ab ≡ (G ba ) −1 .Starting from mutually orthogonal e I ,i and θ a ,I one can construct complementary basis elements e i I and θ I a satisfying (3.29),(3.30)as e i I = G ij e J ,j GJI and θ I a = G IJ θ b ,J G ba .

IV. ONE-LOOP EFFECTIVE ACTION OF RESTRICTED GAUGE THEORY A. Two representations of the one-loop effective action
The reduced representation of the restricted gauge theory, which was constructed above, is subject to the BV formalism of the first-stage reducible model with firststage reducibility generators Z α a ∝ k α b .Therefore, one can use Eq.(2.47) or Eq.(2.52) along with Eqs.(2.48)-(2.50)and the replacement of S[ φ ] by S red [ φ ] in order to build its one-loop effective action.However, as it was discussed above, our goal is to perform quantization in the representation of the original parental fields ϕ I , rather than in terms of the reduced variables φ i .Hence, an interesting task arises -to convert these algorithms into this representation.
Such a conversion is, of course, based on the relations of embedding the restricted theory into the parental one (3.6)-(3.7)and on the classical equation of motion (3.5) which determines the background on top of which the semiclassical expansion is built.These relations imply that all field-dependent entities of the restricted theory, including all gauge-fixing elements, result from the embedding of φ i into the space of ϕ I .Conversely, the objects of the restricted theory are the functions on the parental configuration space.Therefore, like the relation (3.where, to avoid messy notation, we do not supply χ α (ϕ) and other gauge-fixing quantities of the parental theory by hats.In particular, this means that the gauge condition matrices X α ,i , (2.51), and X α ,I in both representations are related by the embedding formula for a covector, The relation between S red ,ij and Ŝ,IJ is trickier.From (3.7) it follows that on shell where we took into account the equation of motion Ŝ,I = λ a θ a ,I and the corollary of Eq. (3.6) θ a ,IJ e I ,i e J ,j + θ a ,I e I ,ij = 0. Therefore, with the inclusion of the gauge-breaking term we have for the operator (2.48) What remains now is to convert the functional determinant of F ij on the space of φ i to the functional determinant of F IJ on the space of ϕ I .This can be done by comparing two expressions for one and the same Gaussian integral with the quadratic part of the action S λ [ ϕ, λ ] in the exponential.On the one hand it equals Dh Dλ e i( 1 where h I denotes perturbations of ϕ I .On the other hand, this integral can be calculated in the parametrization of reduced fields φ i with perturbations h i and specially chosen set of remaining "reducibility" fields θ a with perturbations h a , ) where e I ,i , θ a ,I , e i I , θ I a are calculated on the classical background and chosen to satisfy the biorthogonality relations (3.29), (3.30).Thus, making the change of integration variables h I → (h i (e) , h a (θ) ) we have for the same integral whence the comparison of these two expressions gives For ultralocal restriction constraint (3.2) the reparametrization (4.7) is also ultralocal in spacetime, e I ,i , θ a ,I , e i I , θ I a ∼ δ(x, y), and the last squared determinant here contributes to the effective action δ(0) terms.Repeating the derivation procedure of Eqs.(4.7)-(4.9)with F IJ replaced by some ultralocal symmetric matrix G IJ proportional to undifferentiated delta function of spacetime coordinates -the metric on the configuration space of ϕ I , one finds the expression for Jacobian e I ,i θ I a in the right-hand side of (4.9) in terms of the corresponding functional determinants of this metric, its inverse G IJ ≡ (G J I ) −1 , the induced metric G ij of the reduced φ i -space and the metric G ab in the θ a -directions (see footnote 8) The square of this Jacobian therefore reads (4.12)Note that G ab is ultralocal (for ultralocal restrictions) in contrast to the operator Θ ab defined by Eq. (4.6)Thus, this is an inessential normalization factor of the generating functional or, if treated seriously, it can be absorbed into the definition of the local path integral measure provided one identifies the ultralocal matrix G ij with the Hessian of the gauge-fixed reduced action S red gf [ φ ] with respect to the configuration space velocities φ, 9   Dφ → ,i e J ,j ≡ e I ,i G IJ e J ,j .(4.13) Then det F ij in the one-loop expressions (2.47) or (2.52) get replaced by where raising the indices of operator forms F ij and F IJ is done by the corresponding local metrics of Eq. (4.11) and the second index in the operator of the last determinant is analogously lowered by the "induced" metric G ab along normal directions, (4.15) Replacement of F ij by (4.14) in (2.47) finally gives the one-loop generating functional of a restricted gauge theory fully in terms of the parental theory structures Here the gauge-fixed ghost operator F α β is, of course, given by the unreduced theory version of (2.49) with the projected generators (4.17) 9 Here, of course, the functional dependence of the reduced and parental actions on velocities is mediated by their gauge-fixed Lagrangians, i.e. S red gf = dt L red gf (φ, φ) and S λ gf = dt L λ gf (ϕ, φ).The gauge-fixed action is just the corresponding classical action with added gauge breaking term.Also remember that we consider relativistic gauges, so that the "kinetic" metric (4.13) in reduced theory is nondegenerate.Extending e i I G ij e j J with linear combinations bIaθ a ,J + θ a ,I bJa one acquires the nondegenerate metric GIJ .
Note that despite ultralocal reparametrization (4.7) the transition to parental theory representation leads to the additional nontrivial factor -the determinant of (4.6).The complexity of this factor follows from the fact that this is no longer a determinant of the local differential operator.Rather, this is the determinant of a nonlocal object Θ a b -the Green's function of the local operator F IJ sandwiched between two normal covariant vectors θ a ,I and θ b ,J .

Ambiguity in the choice of generator bases
The BV generating functional (2.10) is not invariant under the change of generator bases of RI α and Z α b as obvious from one-loop expressions (2.47) or (4.16).When linearly transformed with respect to their gauge index α gauge generators RI α still remain the generators of gauge transformations of the parental theory.A unique representations of RI α is not a priori fixed in Lagrangian theory.The linear transformations of Z α b with respect to its reducibility index b are also not fixed and this ambiguity reveals itself in the (yet) arbitrary matrix µ a b (3.27) Ambiguity of the linear reparametrizations of generators results in "normalizing" factors in the generating functional and is the well-known feature of the BV formalism which will be addressed later in this section.Another ambiguity is the choice of a projector parameter k α a which implicitly enters restricted theory gauge generators R I α (3.20) via projectors (3.22) and reducibility generators Z α b ∝ k α a .The only restriction on the choice of k α a is that it should satisfy the rank conditions (3.23).Otherwise it is arbitrary, and this arbitrariness may extend to the effective action (4.16).The problem of potential dependence of the latter on k α a is a specific issue of reducible gauge structure approach to restricted theory and it should be fixed independently of the generators normalization issue.Below we show that the requirement of independence of the generating functional (4.16) on the choice of k α a may be satisfied by a special choice of the factor µ a b in (4.18).

B. Independence of projector parameter
The variational equation can be transformed by using the relation , the analog of the Ward identity (2.60) and the identity (2.56), so that For the first term in the right-hand side of (4.19) this gives where we used the fact that With the expression (4.18) this equation reads in condensed matrix notations where tr denotes the trace over indices of the matrices µ = µ a b and (Qk) = (Qk) a b .Without loss of generality this equation can be solved by µ a b = (Qk) −1 a b , so that finally C. Canonical normalization of generators Specification of the gauge generators basis RI α is a more complicated issue.However, we will give brief and, perhaps, not so exhaustive arguments in favor of a concrete choice which, in particular, will confirm the form (4.24) of Z α b .Conventional Lagrangian quantization in the form of the Faddeev-Popov integral (or the BV integral in more complicated models with open and reducible algebras) is not intrinsically closed.It does not provide uniquely the concrete form of the local measure and does not resolve the associated problem of the choice of the generators bases.In order to fix them one should appeal to the canonical form of the path integral which for a general class of relativistic gauge conditions comprises the BFV formalism.For simple gauge systems10 fixing the measure and the basis of generators looks as follows.
The starting point is the canonical formalism of the gauge theory whose canonical action S can = dt (p q − H − v α γ α ) explicitly contains first-class constraints γ α dual to the Lagrange multipliers v α which are a part of the Lagrangian configuration space of the theory ϕ I = (q, v).This action is invariant with respect to gauge transformations which are canonical (ultralocal in time and generated by Poisson brackets with constraints) in the sector of phase space variables δ ξ (q, p) = {(q, p), γ α }ξ α , but contain the first-order time derivative of the gauge transformation parameter in the sector of Lagrange multipliers, δ ξ v α = ξα + ... .Here dots denote ultralocal in time terms containing structure functions of the Poisson bracket algebra of first-class constraints and the Hamiltonian H, their explicit form being unimportant for us in what follows.The corresponding canonical path integral in the class of canonical gauges χ α = χ α (q, p) reads as and it is obviously invariant under the linear changes of the basis of constraints, γ α → γ ′ α = γ β Ω β α , with any ultralocal in time and nondegenerate matrix Ω β α .Subsequent integration over momenta p converts this integral into the Lagrangian form which, modulo corrections associated with the transition to the Lagrangian expressions for momenta, takes the form of the Faddeev-Popov integral Here the local measure µ[ ϕ ] ∼ exp δ(0)(...) absorbs ultralocal in time factors associated with the above corrections, and the Faddeev-Popov operator Q α β is built in terms of the gauge generators R I β -the Lagrangian version of the above transformations in the canonical formalism, Starting from the BFV canonical quantization 11 one can arrive at the same expression for this wider class of gauge conditions [26,31] including relativistic (or "dynamical") gauges so that gauges χ α involve linear dependence on the time derivatives of Lagrange multipliers v α .The integral (4.26) is not explicitly invariant under the rotation of the generator basis, R I α → R ′ I α = R I β Ω β α , but implicitly the choice of this basis is fixed by the requirement that it should match with the basis of canonical gauge transformations δ ξ ϕ = δ ξ (q, v) implying that δ ξ v α = ξα + ... or R α β = δ α β (d/dt) + ... .The ambiguity of splitting the configuration space of gauge fields ϕ I ∼ (q, v) into canonical coordinates and Lagrange multipliers (which is ultralocal in time) can only lead to some inessential local matrix µ α β in the last equation, ), and some extra factor in the local measure µ[ ϕ ] ∼ exp δ(0)(...) which is again unimportant in concrete applications.Thus, the ambiguity in the choice of generator basis R I β coming from the canonical formalism reduces to its rotation by ultralocal in time matrices, which is equivalent to changing the basis of first-class constraints in the canonical formalism.This ambiguity is physically inessential, so that finally the choice of generator basis is fixed by the requirement of local coefficient of the first-order time derivative in the gauge generators R I α .Unfortunately, in the case of restricted gauge theories such a line of reasoning does not work directly because the reducible generators obtained by the projection procedure may become nonlocal in time.Moreover, the restriction of gauge theory generically leads to its canonical formalism with a much more complicated structure involving many generations of constraints.Therefore it is much harder to implement with the same level of generality the above scheme starting from the canonical quantization.For this reason we will choose a somewhat different approach in order to show that the equivalence to the canonical quantization fixes the basis of projected generators in restricted gauge theory by the requirement that the local parental theory generators RI α are canonically induced, that is they satisfy a local normalization condition for their time derivative part.
Consider a parental gauge theory with the action Ŝ[ ϕ ] whose gauge generators RI α form a closed algebra and are irreducible, so that its generating functional is given by a standard Faddeev-Popov path integral.This integral can be represented on shell (with switched off sources) in three equivalent forms differing by the choice of gaugefixing integration measure, The measure factors M delta (χ) and M(χ,κ) respectively correspond to the delta-function-type and Gaussian-type gauge fixing with the full set of gauge conditions χ α , the latter involving the exponentiated gauge-breaking term with an invertible gauge-fixing matrix The third measure factor is less known and corresponds to the situation when a part of gauge conditions χ p are of delta-function-type, whereas the rest of them are enforced via the gauge breaking term with the projector (4.32)Here is a subset of gauge conditions obtained by projecting the full subset χ α with the aid of a vielbein σ p α = κ pq σ β q κ βα which is dual to the set σ α p introduced above in the formalism of reducible gauge generators, 12 34) On the other hand, inclusion of delta function of the subset of gauge conditions (or partial gauge fixing) can be interpreted as quantization of the restricted gauge theory with partial gauge-fixing conditions χ p playing the role of restriction constraints θ p .According to our derivation above, this restricted gauge theory has reducible gauge generators

RI
α and some set of vectors k α p , so that its path integral over reduced configuration space of φ i should read ) 12 The proof of the last equality in (4.28) can be done by complementing the bases of σ α p with the remaining vielbein vectors σ α M which are orthogonal to σ α p in the metric κ αβ , σ α p κ αβ σ β M = 0 and, therefore, satisfy the determinant relation det ([ , where κpq = σ α p κ αβ σ β q and κ M N = σ α M κ αβ σ β N are respectively the metrics on subspaces spanned by σ α p and σ α M .Then the full delta function of gauge conditions can be decomposed as δ(χ α ) = δ(χ p )δ(χ M )(det κ αβ ) 1/2 /(det κpq det κ M N ) 1/2 and the factor δ(χ M )/(det κ M N ) 1/2 here, according to 't Hooft trick, being replaced by exp(− i 2 χ M κ M N χ N ) = exp(− i 2 χ α Π αβ χ β )the implementation of the second equality of (4.28) in the sector of χ M gauges.
where S red [ φ ] = Ŝ[ ϕ ] | χ p =0 and the gauge-fixing integration measure according to the reducible gauge theory algorithm (2.47) looks like (we distinguish the partial gauge fixing case from the restricted theory case by replacing the indices a, b, ... with the indices p, q, ... from the second part of Latin alphabet)13 F p q = (ωZ) p q = (ωk) p r (Qk) −1 r q , (4.38) where the last equality follows from (4.28).This relation shows that the restricted theory with full-rank operator Q p α is nothing but a partial gauge fixing conceptthe parental and restricted theories which are physically equivalent at the classical level remain equivalent at the quantum level.
The BFV canonical prescription of normalization for gauge generators in the Lagrangian BV approach for the parental theory with the irreducible gauge structure thus imply that correctly normalized reducible "projected" gauge generators R I α = RI β T β α (Q, k) of the restricted (here, partially gauge-fixed) formalism just should be constructed by projecting canonically normalized parental gauge generators RI β .Note that this result also confirms the normalization (4.24) of reducibility generators derived above from another (projector parameter independence) principle.
Let us go over from the case of partial gauge fixing to generic restricted theory physically inequivalent to the parental one.Its path integral in reduced space representation is given by the analog of Eq. (4.35) with the set of restriction conditions θ a instead of χ p , whose operator The transition to integration over the parental theory configuration space, , is written here modulo local measure factors which we will disregard in what follows.The set of functions θ a according to the rank of Q a α can be split into the set of gaugeinvariant functions θ A , range A = m 2 , and the set θ p , range p = m 1 − m 2 , cf.Eqs.(3.12)-(3.13),θ a → (θ A , θ p ) so that where Y = det ∂(θ A , θ p )/∂θ a .The functions θ p enumerated by letters from the second part of Latin alphabet play here the role conditions of partial gauge fixing, χ p ≡ θ p , while the invariants θ A , which are forced to vanish in the path integral, are responsible for inequivalence of the restricted and parental theories.Now, let us choose in the measure M red (χ,κ,σ,ω,k,ρ) the full set of gauge conditions with χ p identified with θ p , χ p = θ p , range p = m 1 − m 2 (cf. the footnote 12 and Eq.(4.33) which applies here in view of orthogonality σ p α σ α M = 0).Let the rest of gauge conditions χ M , range M = m 0 − m 1 + m 2 , form any complementary set of gauges such that the total Faddeev-Popov operator [ Q p β Q M β ] is nondegenerate.Then, substituting the above relation (4.44) in (4.43) and using the relation (4.41) between the measures one has Here, the last equality follows from the equality of measures Eq. (4.28) and the fact that in view of invariance of the Faddeev-Popov deltafunction-type measure with respect to linear transformations of the basis of gauge conditions.Equation (4.46) allows us to make the needed statement: as long as the quantum measure M delta (χ) can be derived from the canonical quantization with canonically normalized local generators RI α , then the same choice of parental theory generators should be used in the construction of restricted theory.
Note that the last expression for Z restricted can be alternatively represented by expressing in terms of original special gauge conditions θ p and χ M and then using (4.44).This allows us to get rid of a potentially nonlocal factor Y and obtain the representation directly in terms of the restriction functions θ a and the complementary subset of gauge conditions, 48) It should be emphasized again that here the Faddeev-Popov operator [ Q p β Q M β ] is built with respect to partial gauge fixing subset of θ a and any complementary to it set of gauge conditions χ M .This recipe may be less suitable in concrete applications, because explicit disentangling the subset θ p from θ a may be involved, so the original form (4.43) should be more useful, and it was explicitly used above for the derivation of the one-loop generating functional (4.16).

D. The difference between parental and restricted theories in the one-loop approximation
For a class of theories with the Jacobian Y independent of integration fields in (4.46) one can write down the representation for Z restricted and its one-loop order (such irrelevant Y will be omitted in the subsequent expressions in this section).In view of gauge invariance of θ A Eq. (4.46) implies on shell the usual Faddeev-Popov integral with κ αβ -fixed gauge and with the gauge-invariant insertion of δ(θ A ) which in the one-loop order by the mechanism of the identical transformation of Eq. (4.5) goes over to a simple relation between the generating functionals of the restricted and parental theories where Ẑ1−loop is obviously the one-loop generating functional of the parental theory given in terms of its (hatted) gauge and ghost inverse propagators and The operator (4.53) is analogous to (4.6), but it is acting in the space of indices A which enumerate gaugeinvariant functions θ A disentangled from the full set of θ a .Note that it is defined in terms of the Green's function of the gauge-fixed operator FIJ of the parental theory with a source λ A (at gauge-invariant observable θ A ).The presence of such source term in parental theory may be interpreted as going off shell and performing one-loop calculations on the family of backgrounds Ŝ,I = λ A θ A ,I , which (together with θ A (ϕ I ) = 0 conditions) specify saddle points of (4.49).This is in accordance with the fact that solutions of these background equations cover all possible backgrounds of the restricted theory.To compare Z 1−loop restricted and Ẑ1−loop in (4.50) these objects of course should be calculated on the same backgrounds. 14 Another important observation confirming the consistency of the relation (4.50) is that the additional factor depending on the matrix Θ AB and this matrix itself are independent on shell of the choice of gauge, δ (χ,κ) Θ AB = 0, which can be checked by using the Ward identities for F −1 IJ derived above.
Finally, let us note on practical aspects regarding the structure of (4.53).Calculation of Θ AB based on nonlocal F −1 IJ may be technically inconvenient.The transition to objects defined in terms of a local differential operator FIJ significantly simplifies calculations.When the dual biorthogonal basis (e I I ′ , θ I A ) satisfying θ A ,I e I I ′ = 0 and θ A ,I θ I B = δ A B , is "orthogonal" with respect to the operator FIJ in the sense that e I I ′ FIJ θ J A = 0, then it is easy to calculate operator Θ BA inverse to Θ AB : Under the above assumptions the determinant (det Θ BA ) −1/2 in (4.50) can be replaced by (det Θ AB ) 1/2 -the inverse operator acquiring simple form in terms of local FIJ V. UNIMODULAR GRAVITY THEORY Application of the above formalism to unimodular gravity theory is straightforward.Its parental theory is Einstein general relativity with the action -the functional of the metric field 14 According to rather generic assumptions on restriction conditions θ a = 0 (see discussion in Sec.III A) the Lagrange multipliers λ a , and thus λ A , are expressible in terms of the background fields.In direct analogy with (3.17) here λA(ϕ) = Ŝ,I (ϕ)θ I A (ϕ) for θ I A (ϕ) being dual to θ A ,I .
where g(x) ≡ − det g µν (x) and R(g µν (x)) is the scalar curvature of this metric (for brevity we work in units with 16πG = 1).The arrow signs ( → ) signify in what follows the realization of condensed notations of previous sections in this concrete field model.Einstein theory is invariant under local gauge transformations δ ξ ϕ I = RI α ξ α -metric diffeomorphisms generated by the vector field ξ α , which read in explicit notations δ ξ g µν = ∇ µ ξ ν + ∇ ν ξ µ , ξ α → ξ α (x) ≡ g αµ ξ µ (x), so that the gauge generators RI α have the form (5.3)By default, derivatives act to the right on the first spacetime-point argument of delta functions.
Unimodular restriction of the theory (5.2) consists in the restriction to the subspace of metrics g µν (x) = g µν (x)/g 1/4 (x) with a unit determinant, g(x) ≡ − det g µν (x) = 1.We will not introduce a special notation for nine independent variables per spacetime point playing the role of φ i but just formulate the restriction constraints θ a (ϕ) as and the gauge-restriction operator (3.11) has the form ) where ∇ α is a covariant derivative with Christoffel connection, here acting on a vector field.
Counterpart of restricted theory (3.4) -the Lagrange multiplier action of unimodular gravity, and its metric equations of motion read The latter resembles Einstein equations with the term mimicking cosmological constant: the Lagrange multiplier λ(x) on shell is indeed constant because according to (3.10) its nonzero part is a spacetime constant zero mode of the gauge-restriction operator (5.6), .9) where the covariant derivative is acting on scalar.This result is obviously equivalent to contracting (5.8) with covariant derivative and using contracted Bianchi identity for the Einstein tensor.On the other hand, tracing this equation one finds λ = R/2, and the set of ten metric equations becomes linearly dependent which corresponds to Eqs. (3.16) and (3.17) of the general formalism of restricted gauge theories, The vacuum solution of these equations is a generic Einstein space metric g µν , R µν = Λ g µν , Λ = λ/2 = const with a unit determinant g ≡ − det g µν = 1.Thus, the left kernel of the operator Q a α of dimensionality m 2 = 1 spanned by the zero mode is what physically distinguishes unimodular gravity theory from Einstein (or Einstein-Hilbert theory with a cosmological constant term) because it allows one to prescribe any constant value of λ from the initial conditions rather than postulate it as a fundamental constant in the Lagrangian of the theory.The relevant gauge-invariant physical degree of freedom constrained by the unimodular restriction according to Eqs. (3.13)-(3.14)above, x which, of course, a priori is not completely specified in Einstein gravity.
The construction of projectors (3.22) with the aid of the matrix k α b satisfying the rank restriction conditions (3.23) suggests the following obvious choice which simultaneously with these conditions provides covariance with respect to spacetime diffeomorphisms, which also provides covariance with respect to spacetime coordinate change.Here ∇ α is the covariant derivative with respect to the dynamical metric g αβ and is its covariant d'Alembertian, which act here on the scalars (we use the standard definition of δ(x, y) as the symmetric kernel of the scalar identity operator), though in what follows we reserve for them the same notation when they will be acting on general tensors and tensor densities.With such choice projector (3.22) reads is a projector on the subspace of spacetime transverse vectors, and it is nonlocal because it is defined in terms of the Green's function of operator, which is understood in the Moore-Penrose sense associated with the rank deficiency of operators (5.6), (5.12), and (5.13)the number m 2 of their spacetime constant zero modes being just 1.

A. Gauge fixing and propagators
Now we go over to the construction of auxiliary elements of gauge-fixing procedure χ α , κ αβ , σ α a , ω a α and ρ a b .To preserve covariance of the formalism we will use background covariant gauge conditions which for simplicity will be linear in the dynamical (quantum) metric field ϕ I = g µν .The coefficient of g µν in χ α , that is χ α ,I , is g µν -independent but explicitly depends on the background metric g µν -the one which on shell satisfies the equations of motion (5.10) and is subject to unimodularity restriction g = 1.Background covariance of such gauge conditions implies that the choice of this coefficient should be such that χ α is covariant with respect to simultaneous diffeomorphisms of both g µν and g µν .Usually as such a gauge one uses the linearized de Donder or DeWitt gauge which is linear in h µν (x) = g µν (x) − g µν (x) -the quantum fluctuation of the metric on top of its background.The DeWitt gauge is , where, as well as below, the covariant derivatives ∇ β and ∇ µ ≡ g µν ∇ ν are constructed in terms of the background metric.In the unimodular case, however, the trace part g µν h µν of the metric fluctuation is systematically projected out, so that it is worth using a simpler gauge For the same reasons of covariance the choice of gaugefixing matrices κ αβ , σ α a , ω a α and ρ a b is also obvious.Just like χ α ,I above we will construct them in terms of the background metric, part of them being directly related to the already introduced quantities κ αβ → κ α,x β,y = g 1/2 g αβ δ(x, y), (5.17) where in (5.18) covariant derivative acts on scalar, whereas in (5.19) covariant derivative acts on vector (forming a covariant divergence).
As the on-shell results do not depend on the choice of these quantities, they could have been chosen in terms of the quantum field g µν , but this would have lead to the origin of extra terms involving functional derivatives κ αβ ,I , σ α a,I , etc. Avoiding such terms essentially simplifies calculations and, particular, allows one to avoid explicit use of the extra ghost C ′a entering (2.21), because in view of (2.22) It should be emphasized that, as long as we restrict ourselves with the one-loop approximation, after all needed functional derivatives have been taken everything gets computed at the background, so that the distinction between g µν and g µν disappears.For this reason we will basically write all the formalism below in terms of the metric g µν with the understanding that it should be restricted to the on-shell unimodular background g µν satisfying (5.10).Objects dual to those defined above, or raising and lowering condensed indices I → (µν, x), α → (α, x), a → x, can be attained by introducing a local configuration space metric G IJ and a similar metric in the space of gauge indices.In Einstein theory a natural choice is the DeWitt metric of the kinetic term of the action, G IJ → G µν,x αβ,y = 1 2 g 1/2 (g µ(α g β)ν − 1 2 g µν g αβ )δ(x, y).In the unimodular context, again due to projecting out the trace part of metric fluctuations, it is more useful to choose G IJ → G µν,x αβ,y = 1 2 g 1/2 g µ(α g β)ν δ(x, y).(5.22)As regards the sector of gauge and reducibility indices, the role of the relevant metrics can be played respectively by κ αβ and κ ab introduced in Sec.II.(Let us remind that by κ αβ we denote the inverse to κ βα , and by κ ba -the inverse to κ ab .) The gauge-fixing choice (5.17) and (5.18) leads to operator (2.40) and dual operators (4.34) which define the projector form (2.43) Partial derivatives acting on delta functions in (5.24) and (5.25) are covariant derivatives ∇ β since they finally act on vector densities (forming a covariant divergence).
With the above objects the construction of the metric and ghost fields inverse propagators (4.4) and (4.17) is straightforward, (5.28) In the derivation of these expressions we used background equations of motion (5.10), so these are essentially onshell objects.In particular, applications beyond one-loop approximation would generate extra terms because of the necessity to distinguish background and quantum fields.
The choice of the gauge-fixing matrix ρ a b in (5.20) allows one to avoid extra term ∇ α ∇ β in the local part of the vector operator F α β (∇).Finally, simplification of the ghosts-for-ghosts operator F a b is due to correct normalization of reducibility generators (4.24) and a special choice (5.19) of ω a α = Q a α .Note that the operators acquired nonlocal parts containing inverse (scalar) d'Alembertians.They were generated due to nonlocal projectors in the gauge-breaking term and in the projected gauge generators.Moreover, even in the local part of the tensor operator covariant derivatives do not form overall d'Alembertian, and their indices get contracted with the indices of test functions on which the operator is acting.This situation is very different from the Einstein theory case for which DeWitt gauge conditions guarantee minimal nature of the operator of field disturbances which is an important property admitting direct use of the heat kernel method for the calculation of the effective action.
The local part of (5.26) can, however, be simplified.Its functional determinant enters the one-loop partition function via the combination (4.9), det F IJ det Θ ab , or (4.14),where Θ ab is defined in terms of F IJ by Eq. (4.6).It is easy to check that this combination is invariant under the change of the operator F IJ of the form because Therefore, we can omit in (5.26) the terms of the second line which are proportional to θ a ,I ∼ g µν and θ a ,J ∼ g αβ and replace this operator by (5.31) accordingly taking the operator Θab induced by F −1 IJ = F −1 µν αβ (∇) δ(x, y), Still, all the vector (5.27), tensor (5.31) and scalar (5.33) operators remain nonlocal, and their remains a problem of reducing their determinants to some calculable form.

B. Reduction of functional determinants
Reduction of the above determinants can be done by the decomposition of the space of tensor and vector fields into irreducible transverse and traceless components.However, this decomposition in concrete applications can be useful only when the underlying metric background is homogeneous and the bases of irreducible scalar, transverse vector and tensor harmonics with their explicit spectra are known.If one wants to work on generic backgrounds and use such general methods as heat kernel method or Schwinger-DeWitt technique of curvature expansion [24], then the above operators should be transformed to the form of differential or pseudo differential operators with simple principal symbols, preferably local and minimal ones that are constructed of covariant derivatives which form powers of covariant d'Alembertians.This is especially important for the operator (5.33) having essentially nonlocal structure with the Green's function of another nonlocal operator.
In order to reduce the calculation of the determinant of F µν αβ (∇) to that of the minimal operator let us include it into the one-parameter family (5.34) interpolating between F µν αβ (1|∇) = F µν αβ (∇) and the local minimal operator F µν αβ (0|∇) = ∆ µν αβ (∇), where the double derivative of the inverse operator F −1 µν γδ (a|∇) can be obtained by the following sequence of transformations.When applied to F µν αβ (a|∇) this double derivative reads as a local expression where we used the fact that ∇ µ R µανβ = 0 on Einstein background.Functionally contracting this relation with the inverse of F µν αβ (a|∇) on the right and with the inverse of the scalar operator g 1/2 2(1 − 2a) + (1 − a)R on the left, we obtain where we took into account that on Einstein manifold when this operator is acting on a scalar.Substituting this result in (5.36), using cyclic permutation under the trace and integrating over a from 0 to 1 we have Tr ln F µν αβ (1|∇) = Tr ln F µν αβ (0|∇) (5.40) Here Tr ′ implies taking the functional trace of the operator over the space of eigenmodes of the scalar operator excluding its constant zero mode.The explanation of this important fact follows from the observation that the action of the operator (5.39) on a covariantly constant mode obviously gives zero in view of the positive power of in the numerator.Therefore, even the multiplication by 1/ in (5.36) does not make the contribution of this mode in the functional trace nonzero; this is obviously consistent with the Penrose-Moore prescription for the inverse of discussed above.
With the inclusion of the local measure factor (5.40) then becomes where the prime in det ′ obviously implies the same rule -omission of the zero eigenvalue of in the definition of the functional determinant, and this of course refers to both the numerator and denominator of the operator valued fraction under the sign of det ′ .Similar steps for the vector operator (5.27) on the Einstein metric background result in (5.42) The reduction of the determinant of the scalar operator Θ(∇), which is defined by Eq. (5.33) and does not at all have a local part, can also be done via the transformations of the above type.First of all consider the contraction which localizes the operator F µν αβ (∇), (5.43) Contracting this relation with F −1 αβ µν (∇) and using Eq.(5.38) with the parameter a = 1 we get so that the operator (5.33) and its determinant take the form (5.45) Assembling together in the tilde version of (4.16) the results (5.41), (5.42), (5.45), trivial contribution of det ρ = 1 and noting that det κ ab = det ′ we finally get equals the above minimal operator ∆ αβ µν (∇) and the Faddeev-Popov ghost operator Q α β = δ α β + R α β also coincides with the vector operator ( + 1 4 R)δ α β .So if we consider Einstein gravity with the cosmological constant as a parental theory of unimodular gravity then the relation (5.46) can be interpreted as Eq.(4.50) relating the generating functionals of the restricted theory and the parental one with Ẑ1−loop → Ẑ1−loop E (Λ), provided we can prove equality of factors (5.47) and (det Θ AB ) −1/2 .This proof is straightforward.
The only invariant θ A that can be built out of the restriction function (5.4)-(5.5)by integrating it with the constant zero mode of the gauge-restriction operator is the following global quantity which reads along with its θ A ,I as x (g 1/2 (x) − 1), (5.51) θ A ,I → θµν (y) = 1 2 g 1/2 g µν (y). (5.52) In order to find its Θ AB , ΘAB → Θ ≡ d 4 x d 4 y θµν (x) F −1 µν αβ (∇)δ(x, y) θαβ (y), (5.53) we need the gauge field inverse propagator of the parental theory -Einstein gravity with a cosmological constant, which reads on shell as This operator satisfies an obvious relation which allows one to find the following contraction of the Green's function kernel with metric tensors (5.56) whence the needed factor (5.53) equals (5.57) where the last equality follows from the uniformity of R = 4Λ in spacetime.Of course on shell, g µν = g µν , we have g 1/2 = 1, so the square root of the metric determinant is retained entirely for the sake of manifest covariance, and in this way it represents invariant volume of spacetime.Thus, (5.58) which up to a constant factor coincides with (5.47). 15or completeness we present here the same answer rewritten in the basis of irreducible subspaces of tensor and vector fields, which are defined via disentangling from the full tensor field its tranverse-traceless h TT = h TT µν , transversal vector h T = h T µ and two scalar parts [32], (5.59) Similarly for a vector such a decomposition reads ) in the functional integration over h µν and v µ equal , (5.61) Here det ′ is a functional determinant of a scalar operator with omitted zero mode of the -operator (note that this is the omission of the (constant) zero mode of , but not the zero mode of the operator whose determinant is being taken). 16Similarly det T denotes the determinant of the vector operator taken on the space of transverse vector functions.Quadratic forms with tensor field ∆ µν αβ (∇) and vector field ( + R/4)g µν kernels correspondingly read Taking the Gaussian integrals with these quadratic forms and using the above Jacobians one can find the representation for the determinants of Eq. (5.46) in terms of their irreducible counterpart -transverse-traceless tensor det TT and transverse vector det T ones, (5.66) 16 The omission of these spacetime constant zero modes of takes place because they do not contribute to the left-hand sides of Eqs.(5.59)-(5.60).
whence the partition function in Einstein theory with the cosmological constant (5.49) reads (5.67) In terms of these determinants on irreducible subspaces of transverse-traceless modes it differs by extra factor from the usually claimed form [33].This factor originates on account of a constant zero mode of a scalar d'Alembertian.It should be emphasized that on homogeneous de Sitter or anti-de Sitter background other vector and tensor operators also have zero modes associated with Killing symmetries of these backgrounds (see e.g.[33]).Here we disregard them because we consider generic inhomogeneous Einstein metric spacetimes for all of which this zero mode of always exists.
Thus, in terms of determinants on constrained (irreducible) fields the one-loop result for Einstein theory with the cosmological constant differs from a conventional expression by the contribution of one constant eigenmode of the operator + R 2 .Curiously, for unimodular gravity this contribution in the same representation completely cancels by additional factor in (5.46), and we have This result coincides with the one claimed in [13,15].It manifestly exhibits the counting of local physical degrees of freedom -5 traceless-tensor modes minus 3 transverse vector modes.

VI. CONCLUSION
To summarize our results, we worked out a full set of gauge-fixing elements in generic gauge theory of the first-stage reducibility and constructed a workable algorithm for its one-loop effective action.We also derived the set of tree-level Ward identities for gauge field, ghost and ghosts-for-ghosts propagators, which allow one to prove on-shell gauge independence of the effective action from the choice of auxiliary elements of gauge-fixing procedure.We showed that Lagrangian quantization of a restricted theory originating from its parental gauge theory can be performed within the BV formalism for models with linearly-dependent gauge generators of the first-stage reducibility.It turns out that new physics contained in the restricted theory as compared to its parental theory model is associated with the rank deficiency of a special gauge-restriction operator reflecting the gauge transformation properties of the restriction constraints functions.The choice of first-stage reducibility generators, or zero vectors of the projected gauge generators induced from the parental theory, has a certain freedom limited only by a special rank restriction condition, but the on-shell independence of physical results from this choice is provided by a special normalization of these vectors.
These general results are applied to the quantization of unimodular gravity theory.Its one-loop effective action, initially obtained in terms of complicated nonlocal pseudodifferential operators, is transformed to functional determinants of minimal second-order differential operators calculable on generic backgrounds by Schwinger-DeWitt technique of local curvature expansion.This also confirms the known representation of one-loop contribution in unimodular gravity theory in terms of functional determinants on irreducible transverse and transverse traceless subspaces of tensor, vector, and scalar modes.The oneloop order in unimodular gravity turns out to be equivalent to that of Einstein gravity theory with a cosmological term only up to a special contribution of the global degree of freedom associated with the variable value of the cosmological constant.
From the viewpoint of local phenomenology, the new physics in unimodular gravity turns out to be of a somewhat borderline nature.Classically it is manifested in the fact that the cosmological constant in UMG becomes a part of initial conditions rather than a fundamental constant in the Lagrangian of the Einstein theory.At the one-loop level the contribution of this extra global and spacetime constant degree of freedom is very peculiar, and the way it shows up depends on the representation of the theory.In fact we have two somewhat complementary representations for both theories: in terms of functional determinants of differential operators on full field spaces or on spaces constrained by irreducible representations.The one-loop Einstein gravity in the full space representation (5.49) does not reveal the contribution of this mode, whereas the constrained determinants representation (5.67) makes it manifest.With two representations for unimodular gravity (5.46) and (5.68) this situation is reversed.
The manifestation of this contribution is the powerlike dependence on (5.57) in the partition function ∼ d 4 x g 1/2 /Λ which becomes in the effective action a logarithmic essentially nonlocal contribution.Off shell, that is in transition to gradient expansion for nonconstant curvature scalar, R → R(x), it may go over into the structures like ln d 4 x g 1/2 (x) 1/R(x) + O(∇R) .These structures might be important in Euclidean quantum gravity responsible for tunneling phenomena and gravitational thermodynamics.In fact, thermodynamics reveals the duality relation between Einstein theory and unimodular gravity as the analogy of the Laplace transform relating the statistical ensemble with fixed volume vs the fixed pressure ensemble.Qualitatively, this can be shown as follows.
By identifying in (5.11) the coordinate 4-volume d 4 x = V as the fixed argument of the generating functional (4.49) we have the expression for this functional in UMG theory (in Euclidean picture this is a partition function at fixed volume V ) where the measure M incorporates, just like in (4.49), the full local gauge fixing of all four dimensional diffeomorphisms.Then, it is obvious that the Laplace transform with respect to the volume variable converts the UMG partition function Z UMG (V ) to that of the Einstein theory Z E (Λ) with a fixed value of Λ -the cosmological constant dual to V , (here we reinsert the gravitational constant factor 1/16πG and note the opposite sign of the Euclidean version of the action (5.48)).Of course, this derivation should be regulated by specifying the boundary conditions which fully determine a finite value of V (or its infinite limit) and a finite value of the action, achieved by a subtraction of proper surface terms.This can be done along the lines of [11], but in the present form it already conveys the essence of duality between Einstein theory and UMG gravity.
It should be emphasized that throughout our derivations we used Moore-Penrose concept of inverting operators with zero modes, which by and large corresponds to the omission of the zero-mode subspace.This makes all derivations, as discussed in Appendix A, consistent, but apparently leaves room for nontrivial effects of the above extra contributions based on a careful treatment of boundary conditions.Despite the fact that the functional determinants of the scalar d'Alembertian in Eqs.(5.61)-(5.67),which are vulnerable to zero-mode treatment, completely cancel out, there still may be a subtlety in their treatment and this might amount to the extension of the BV method beyond first-stage reducibility.Note that m 2 -dimensional zero-mode subspace of gaugerestriction operator (3.11) is exactly the playground for second-stage reducibility in the general formalism.The smallness of the phase-space sector of this mode in UMG, m 2 = 1, does not make it less important and might be at the core of cosmological constant problem.All this, however, goes beyond the scope of this paper and remains a subject of further research to be reported elsewhere.
Another direction of further research might be the generalized unimodular gravity (GUMG) of [6,7], which is interesting in view of its dark energy and inflation theory implications.This model is more complicated than UMG, it has more complicated canonical formalism encumbered by the presence of second-class Dirac constraints and it strongly breaks diffeomorphism and Lorentz symmetry because of replacement of the UMG restriction condition det g µν = −1 by the Lorentz noninvariant relation between the lapse function and spatial metric.New physics in this model is associated with the origin of the dark perfect fluid which might serve as a source of dark energy or play the role of inflaton, i.e. scalar graviton degree of freedom [8].Covariant quantization of this model along the lines of the BV method applied to parental Einstein gravity is also a good nontrivial playground for our technique.The second group of terms in the right-hand side of (B1) requires differentiation of the projector T γ α (3.22).This projector involves the procedure of inverting the matrix (Qk) a b whose rank is lower than the range of its indices and, therefore, requires the Moore-Penrose construction of the generalized matrix inversion [25].This in turn leads to subtleties of variational procedure for (Qk) −1 a b discussed in Appendix A. As shown there, the variational property of the projector is effectively equivalent to the naive use of the variational rule δ(Qk) −1 = −(Qk) −1 δ(Qk) (Qk) −1 , provided the rank restriction condition (3.23) holds, and it reads as (A9).This can be directly applied to the second group of terms in the right-hand side of (B1) on using the symmetry θ b ,IJ = θ b ,JI , the algebra of parental theory generators, and their corollary Q Here N γ αβ is defined by Eq. (3.37) and L α β is the longitudinal projector complementary to transverse projector T α β , Therefore, summing the contributions (B2) and (B3) one observes that their first terms form out of RI ǫ the projected generator R I ǫ , so that we get  It is also assumed that all determinants in (C3) are nonzero.
Note that the m 0 × m 0 matrix in the first determinant in the right-hand side of (C3) can be written as = ÉÌ + σ a, Ì = Á − k (σÉk) −1 σ É, (C4) where Ì ↔ T α β (σÉ, k) is in fact the oblique projector (3.22) with the left kernel σ É ↔ σ a β Qβ α = Q a α which can be identified with the gauge-restriction operator.The first term of the matrix is therefore degenerate and has as right and left zero vectors k and σ respectively, ÉÌ k = σ ÉÌ = 0. Therefore it can be interpreted as the analog of the Hessian of a gauge-invariant action, whereas the second term in plays the role of a gaugefixing term providing invertibility of this matrix.
The proof of the relation (C3) can be done by using the basis in which the matrix acquires a block-triangular form, but it is easier to use the analog of Ward identities in order to prove that the right-hand side is actually independent of the choice of arbitrary elements σ, a and k and then check that this relation indeed holds under a special choice of these elements.Such a choice is obvious and reads as σ = Ék and a = σ É (when σ Ék = a k = I), so that it remains to check that the right-hand side is indeed (σ, a, k)-independent.
Multiplying the matrix (C4) from the left and from the right by zero vectors of its first term ÉÌ one finds two Ward identities and their corollary, Then, direct variation of the right-hand side of the relation (C3) with respect to a, k and σ shows that it is indeed independent of these quantities on account of the above identities and the variational version of duality relation (C2), σ δσ + δσ σ = 0. We finish this discussion with the note on applicability of the above relation to the case when the objects labeled by indices a are rank-deficient (with the rank m 1 − m 2 < m 1 ).In Sec.IV C, where this determinant relation was used, instead of indices a we explicitly used the indices p belonging to the range m 1 − m 2 which symbolized the maximal rank (irreducible) representation.In such a representation all matrices are nondegenerate and quantities with mixed indices are of the maximal rank.However the maximal rank representation is not necessary and this proof runs equally well in the reducible representation of rank-deficient objects (σ, σ, a, k) of rank m 1 − m 2 , provided the composite quantities σ a, (ak), (σÉk) have the same rank m 1 − m 2 20 and the inverse degenerate matrices (σÉk) −1 , (ak) −1 are treated in the Moore-Penrose sense (as was discussed in Appendix A).Determinants of degenerate operators of rank m 1 − m 2 should be understood in an appropriate regularized sense, for example by omitting the contribution of zero eigenvalues in the eigenvector bases.If one goes into details of the variational proof, then the variations of regularized determinants turn out to be δ det (σÉk) = tr δ(σÉk)(σÉk) −1 with the Moore-Penrose definition of the inverse matrix.The uniqueness of Ì projector variation was shown in Appendix A. The uniqueness of the regularized determinant variation has a similar mechanism.This line of reasoning justifies the validity of one-loop contributions to the effective action of the restricted theory of Secs.III and IV A (even though in Sec.II these contributions were formally based on full-rank quantities with reducibility indices a).

action 8 III. Reducible gauge structure of restricted gauge theories 9 A
action for theories with first-stage reducible gauge generators 3 A. Batalin-Vilkovisky formalism for reducible gauge theories 3 B. Stationary point of the gauge-fixed master action 6 C. One-loop contribution to the generating functional 8 D. Ward identities and gauge independence of the effective .Two representations of a restricted theory 9 B. Gauge symmetry and reducibility 11 C. Gauge algebra of a restricted theory 12 ) which are constrained to vanish.So generically the restricted theory(3.4) is inequivalent to the parental gauge theory Ŝ[ ϕ I ], because the latter does not a priori impose any restrictions on its gauge-invariant objects.In what follows we will consider such restricted theories which incorporate new physics beyond their parental ones.There is another representation of the solution for Lagrange multipliers.If one constructs the set θ I b dual to covariant vectors θ a ,I ,θ a ,I θ I b = δ a b ,(3.15) then, contracting the equation of motion (3.5) with θ I b one obtains 20) and Raα = 0 is enough for the purpose of this paper.In generic case when one uses the projector(3.22)with some other constant rank left kernel,Q ′a α = Q a α + Γ a bα θ b , differing from Q a α(3.11) off the restriction surface, Noether identities imply nonzero generators R bα = λaΓ a bβ T β α (Q ′ , k).This is inevitable when the rank of Q a α (3.11) jumps outside of the restriction surface.
27) with arbitrary factor µ b a , so that rank k β b µ b a = rank k β b .Such natural choice imply off-shell first-stage reducibility (3.26) of projected gauge generators.

4 ,
numerical, but in fact it is a function of the dynamical global degree of freedom Λ belonging in UMG to the full configuration space of the theory.Modulo this extra factor, the result (5.46) exactly coincides with the one-loop contribution of gravitons in Einstein theory with the actionS Λ [ g µν ] = d 4 x g 1/2 (R − 2Λ)(5.48)and the on-shell value of the cosmological constant Λ = R/on-shell inverse propagator of this model in the De-Witt gauge, in view of the open algebra relation for RI α the first group of terms here reads RI γ T γ ǫ k ǫ a,J (Qk) −1a b Q b Ŝ,J T γ α T δ β (B5)where D I K is defined by Eq. (3.38),DI K ≡ δ I K − RI ǫ k ǫ a (Qk) −1a b θ b ,K ,(B6)and we used the fact thatR I ǫ = R I β T β ǫ , k α a = L α β k β a and Q b γ = Q b β L β γ .Here the last term is not explicitly antisymmetric in indices I and J.However, due to the property Ŝ,J = Ŝ,I D I J of the projector D I J it can equivalently be rewritten in the antisymmetric form withE IJ αβ = D I K D J L ÊKL γ δ T γ α T δ β .This finally leads to the algebra (3.34) with structure functions (3.35)-(3.38).The oblique projector D I J defined by (3.38) enters the formalism when the parental algebra is open.Its left kernel is spanned by (Qk) −1a b θ b ,I and the right kernel is spanned by the set of longitudinal gauge vectors RJ β k projector in the ϕ I -space converts the parental generator into the projected one in the space of gauge indices, D I J RJ β = RI α T α β ≡ R I β , so that θ b ,I D I J RJ β = 0.When rank θ a ,I = rank Q a α = m 1 then the second of relations (B7) can be simplified to θ b ,I D I J = 0.For rank-deficient Q a α , when rank θ a ,I = m 1 and rank Q a α = m 1 − m 2 , the correct property is θ b ,I D I J = 1−(Qk)(Qk) −1 b a θ a ,I ≡ L 1 b a θ a ,I , where L 1 b a is a projector on left zero vectors of Q a α .For this generic (physically interesting case) the rank deficiency of D I J equals the rank of gauge-restriction operator (3.11), corank D I J = rank Q a β = rank k α b .Note that the resulting open algebra of the restricted theory (3.34) closes on shell of the parental theory Ŝ,I = 0 rather than on its own shell Ŝ,I − λ a θ a ,I = 0.In view of the above relations one has Ŝ,I D I J = ( Ŝ,I − λ a θ a ,I )D I J + λ a L 1 a b θ b,J , so that the algebra (3.34) can be rewritten in the formR I α,J R J β − R I β,J R J α = R I γ C γ αβ +E IJ αβ Ŝ,J − λ a θ a ,J + E IJ αβ λ a L 1 a b θ b ,J ,(B8) where the last term, unless it is zero, breaks its closure on shell of the restricted theory.Note that λ a L 1 a b is the part of the full set of Lagrange multipliers λ a which stay unrestricted by the equation of motion (3.5) for Lagrange multipliers.

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Appendix C: Determinant relationIn this section we prove the determinant relation(4.40).All involved quantities are matrix (two-index) structures which allow us to omit indices implying standard matrix multiplication.There are indices of two types -lowercase Greek indices of some range m 0 and lowercase Roman indices of the lower range m 1 < m 0 Regarding the rank of quantities with indices a, b, .. see a brief discussion in the end of this section.Quadratic m 0 × m 0 matrices are denoted by capital Roman letters (e.g.Ì).The matrices mapping from m 0dimensional space to a lower m 1 -dimensional space and back will be either underlined or will respectively carry a line over them.Thus we haveÉ ↔ Qα β , Á ↔ δ α β , a ↔ a a α , σ ↔ σ a α , σ ↔ σ α a , k ↔ k α a , σ a ↔ (σ a) α β ≡ σ α a a a β , σ Ék ↔ (σÉk) a b ≡ σ a α Qα β k β b , a k ↔ (a k) a b ≡ a a α k α b .(C1)Notethat the matrix É here is a quadratic m 0 × m 0 matrix and in Sec.IV C it stands for the complete Faddeev-Popov operator for the rank-m 0 parental gauge symmetry.We also assume that σ and σ form dual vectors in the sense thatσ σ = I ↔ σ a α σ α b = δ a b .(C2)Then one can prove the following relation between the determinants of these matrices detÉ = det(σÉk) det(ak)× det É Á − k (σÉk) −1 σ É + σ a .(C3)